On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C _M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C _M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C _M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

Semiprime Goldie centralizers

LetG be a finite group of automorphisms acting on a ringR, andR ^G={fixed points ofG}. We show that under certain conditions onR andG, whenR ^Gis semiprime Goldie then so isR. In particular, ifa∈R is invertible anda ⁿ∈Z(R), thenR ^G,withG generated by the inner automorphism determined bya, is the centralizer ofa—C _R(a). The above result withR ^Greplaced byC _R(a) is shown without the assumption thata is invertible.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

Some results on the cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module Ext _R ^t (R/a,M) is finite (finitely generated), H _a ⁱ (M) is a-cofinite for all i < t and H _a ^t (M) is minimax then H _a ^t (M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and H _a ⁱ (N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module H _a ^t (M,N) is finite.     

The biweight enumerator of self-orthogonal binary codes

Let C be a binary code of length n and let J_C (a, b, c, d) be its biweight enumerator. If n is even and C is self-dual, then J_C is an element of the ring R^$\text{G}$ of absolute invariants of a certain group $\text{G}$. Under the additional assumption that all codewords of C have weight divisible by 4, a similar result holds with a different group. If n is odd and C is maximal self-orthogonal, then J_C is an element of a certain R^$\text{G}$-module. Again a similar result holds if the codewords of C have weights divisible by 4. The groups involved are related to finite groups generated by reflections. In this paper the structure of these groups is described, and polynomial bases for the rings and modules in question are obtained. This answers a question posed in The Theory of Error- correcting Codes by F.J. MacWilliams and N.J.A. Sloane.     

Relations between the half turns of the hyperbolic plane

Let R_a denote the half turn about the point a of the hyperbolic plane H. If the points a, b, c, d lie on the same line and the pair (c, d) is obtained from the pair (a, b) by a translation, then we have R_aR_b = R_cR_d. We study the group G whose generating set is {R_a:a∈H} and whose defining relations are the ones mentioned above together with the relations R²_a = 1. We show that G can be made into a Lie group, G has two connected components, and its identity component G₀ is the universal covering group of PSL₂(R). In particular, it follows that all relations between the half turns in PSL₂(R) follow from the abovementioned relations and a single additional relation of length five.     

Almost fixed-point-free automorphisms of order 2

Let φ be an automorphism of order 2 of the group G with C _G (φ) finite. We prove the following. If G has finite Hirsch number then G is (nilpotent of class at most 2)-by-finite but need not be abelian-by-finite. If G is a finite extension of a soluble group with finite abelian ranks, then G is abelian-by-finite.     

Characterization of Gromov hyperbolic short graphs

To decide when a graph is Gromov hyperbolic is,in general,a very hard problem.In this paper,we solve this problem for the set of short graphs(in an informal way,a graph G is r-short if the shortcuts in the cycles of G have length less than r):an r-short graph G is hyperbolic if and only if S9r(G)is finite,where SR(G):=sup{L(C):C is an R-isometric cycle in G}and we say that a cycle C is R-isometric if dC(x,y)≤dG(x,y)+R for every x,y∈C.     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

Products of powers in finite simple groups

LetG be a group. For a natural numberd≥1 letG ^d denote the subgroup ofG generated by all powersa ^d ,a∈G. A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromG ^d can be represented asa ₁ ^d ...a _N ^d ,a _i ∈G. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsG ^d are closed and that an arbitrary subgroup of finite index inG is closed. In [5,6] the first author proved the existence of such a function for nilpotent groups and for finite solvable groups of bounded Fitting height. Another interpretation of the existence ofN(m, d) is definability of power subgroupsG ^d (see [10]). In this paper we address the question for finite simple groups. All finite simple groups are known to be 2-generated. Thus, we prove the following: THEOREM:There exists a function N=N(d) such that for an arbitrary finite simple group G either G ^d =1 orG={a ₁ ^d ...a _N ^d |a _i ∈G}. The proof is based on the Classification of finite simple groups and sometimes resorts to a case-by-case analysis.     

Some results about the cross-correlation function between two maximal linear sequences

Let {a₁} and {a_d1} be two maximal linear sequences of period pⁿ ? 1. The cross-correlation function is defined by C_d(t) =

for t = 0, t…pⁿ ? 2, where $\text{ζ = exp(2π}\text{1}\text{p}\text{)}$. We find some new general results about C_d(t). We also determine the values and the number of occurences of each value of C_d(t) for several new values of d.     

Piecewise Endomorphisms of Pid-Modules

Let G be a finitely generated module over a PID, D. We investigate the structure of the centralizer near-ring M_D(G) = {f: G → G ¦ f(ar) = (fa)r,a ∈ G, r ∈ D}. If C = {G_α} is a cover of G by maximal cyclic submodules then we show that every f ∈ M_D(G) is piecewise an endomorphism of G.     

The Atiyah bundle and connections on a principal bundle

Let M be a C ^∞ manifold and G a Lie a group. Let E _G be a C ^∞ principal G-bundle over M. There is a fiber bundle C(E _G ) over M whose smooth sections correspond to the connections on E _G . The pull back of E _G to C(E _G ) has a tautological connection. We investigate the curvature of this tautological connection.     

Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients

In this paper we consider nonautonomous elliptic operators A with nontrivial potential term defined in I×Rd, where I is a right-halfline (possibly I=R). We prove that we can associate an evolution operator (G(t,s)) with A in the space of all bounded and continuous functions on Rd. We also study the compactness properties of the operator G(t,s). Finally, we provide sufficient conditions guaranteeing that each operator G(t,s) preserves the usual Lp-spaces and C₀(Rd).     

On some spherical t-designs

Let G be a finite subgroup of the orthogonal group O(d). It is shown that many spherical t-designs are constructed from G, if some particular irreducible representations of O(d) remain irreducible when restricted to G.     

Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let ${(R, \mathfrak{m})}$ be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of ${R, H _{\mathfrak{m}} ^{d} (R)}$ , has finite G _C -injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

Subgroups of prescribed finite index in linear groups

LetR be a commutative ring,M a finitely generatedR-module andG a subgroup of Aut _R M. Under either of the following conditions, for every positive integerd there is a normal subgroupH ofG of finite index such thatG/H contains an element of orderd. (a)G is infinite and finitely generated. (b)R is finitely generated as a ring andG is not unipotent-by-finite. This extends recent work of A. Lubotzky.